Defensive Strategies in the Quality Ladders

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(1)DOCUMENT DE TRAVAIL. DT/2010-11. Defensive Strategies in the Quality Ladders. Ivan LEDEZMA. UMR DIAL 225. Place du Maréchal de Lattre de Tassigny 75775 • Paris Cedex 16 •Tél. (33) 01 44 05 45 42 • Fax (33) 01 44 05 45 45 • 4, rue d’Enghien • 75010 Paris • Tél. (33) 01 53 24 14 50 • Fax (33) 01 53 24 14 51 E-mail : dial@dial.prd.fr • Site : www.dial.prd.fr.

(2) DEFENSIVE STRATEGIES IN THE QUALITY LADDERS1 Ivan Ledezma2 Université Paris Dauphine, LEDa UMR 225 DIAL, IRD ivan.ledezma@dauphine.fr Document de travail UMR DIAL Septembre 2010 Abstract This paper analyses the potentially defensive behaviour of patent race winners and its effect on aggregate R&D effort. It proposes a quality-ladders model that endogenously determines leaders technology advantages and who innovates. Product market regulation can have either a positive or a negative effect on R&D intensity. The negative effect is likely to be observed in highly deregulated economies. The positive influence arises in more regulated environments and it is stronger for larger innovative jumps. These steady-state equilibrium outcomes are consistent with puzzling and contrasting patterns stemming from data on manufacturing industries for 14 OECD countries during the period 1987-2003. Key words : Innovative leaders, quality ladders, product market regulation, R&D. Résumé Dans cet article nous étudions le comportement potentiellement défensif des innovateurs et son effet sur l’effort agrégé d’innovation. Un modèle à échelles de qualité est proposé afin d’analyser l’émergence d’avantages technologiques qui, in fine, déterminent qui innove (le leader ou ses concurrents). Dans ce contexte, la réglementation de marché peut avoir un effet positif ou négatif sur l’intensité en R&D. Elle peut être négativement associée à l’effort d’innovation dans des environnements hautement dérèglementés. Par contre, en économies qui dépassent un certain seuil de réglementation, susceptible de limiter effectivement la construction de barrières stratégiques, la réglementation induit des incitations à innover. Ces prédictions sont cohérentes avec des tests empiriques menés sur un échantillon d’industries appartenant à 14 pays de l’OCDE durant la période 1987-2003. Mots clés : Leaders innovants, modèle à échelles de qualité, réglementation, R&D. JEL Classification : L13, O31, O33.. 1. 2. The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2011) under grant agreement n°225349 (ICaTSEM project). I am grateful to Bruno Amable, Philippe Askenazy, José Miguel Benavente and Dominique Guellec for their helpful and detailed comments. I have also benefited from the reading and suggestions of Maria Bas and Elvire Guillaud. All remaining errors, of course, are my own. I started this work as a research fellow at the Centre d’Economie de la Sorbonne..

(3) 1. Introduction. Several empirical studies based on R&D surveys show that …rms protect the value of their innovations using multiple strategies (Levin et al., 1987; Nelson and Walsh, 2000; Cohen et al., 2002). It is argued in this paper that this multiplicity is important to understand the e¤ect of product market regulation on R&D incentives. If …rms have several alternatives to keep their pro…ts, potential competition may not necessarily act as a slack-reducing device. Rather than neutral innovative behaviour, the thread of competition can in practice trigger defensive reactions of incumbents. They can construct di¤erent types of strategic barriers aiming at protecting their business position from the risk of loosing future innovation contests.1 This paper analyses the channels through which product market regulation, by setting the boundaries of …rm strategies, can in‡uence aggregate R&D e¤ort. A key motivation comes from contrasted empirical patterns characterising the link between regulation and R&D e¤ort. Industry-level data on a sample of OECD manufacturing industries for 1987-2003, a period marked by important market reforms, reveals that in markets with a low level of regulation, the intensi…cation of regulation is negatively associated with R&D intensity. However, the opposite is true in highly regulated environments. Moreover, the positive link between regulation and R&D intensity observed therein is all the more important that manufacturing activities relate to the production or to the use of ICT goods (high-tech industries). These patterns are robust to several tests and, as surprising as it may be, they are broadly consistent with the existing empirical literature. In order to explain these …ndings the paper proposes a quality-ladders setting that emphasises the role of strategic behaviour in the process of innovation. R&D races are structured by the erection of strategic barriers, the cost of which is assumed to be positively correlated with regulation. Within a Stackelberg game of the kind presented in Barro and Sala-i-Martin (2004), an important contribution of the model is that the new successful innovator strategically acquires R&D cost advantages vis-à-vis his competitors. These advantages allow him to further innovate. The main result is that regulation can have either a positive or a negative e¤ect on R&D intensity. It all depends on the pre-existing regulation level. In more liberal environments the equilibrium is characterised by a long-life innovative monopolist. Here an increase in regulation will be detrimental to innovation because it distorts the innovative activity of the leader. However, after attaining a certain threshold of regulation, the economy experiments Schumpeterian replacement. In this case, regulatory provisions can positively in‡uence aggregate R&D e¤ort since they reduce the deterring e¤ect on outsiders. Interestingly, because of Schumpeterian incentives stemming from the technology gap between leaders and followers, this positive impact is all the more important that the size of innovation is larger. Innovation in hightech industries usually leads to more important technology upgrades than in the rest of industries. Hence, this prediction is also consistent with the above-mentioned di¤erentiated e¤ect of regulation on those industries. Moreover, the fact that schumpeterian replacement only arises after a certain level of regulation and that the presence of a permanent monopolist hinds a constant menace of competition, highlights the complexity of the link between product market regulation and product market competition, that is to say between de jure and de facto dimensions of market structure.2 1. Consistent with R&D surveys …ndings, Crépon and Duguet (1997) show evidence of negative R&D externalities among French manufacturing …rms in narrowly de…ned industries, a result interpreted by the authors as the outcome of competitors’rivalry. 2 The Economist in the title of the printed article of May 1st 2008 re‡ects the non-trivial link between regulation. 2.

(4) Regulation is usually modeled as a …xed entry cost without other consequence than the misallocation of resources. By contrast, here regulation not only involves an entry cost but also acts as a device constraining the set of available strategies, namely those leading to entry deterrence in R&D. In the model this kind of strategies are illustrated thanks to a multidimensional representation of quality. In order to fend o¤ followers, the leader can introduce additional complexity into the new version of the good and change the direction of the innovative path. The idea is that innovation is not only a purely product/process improvement but a strategically “biased” one. Pro-competitive provisions regulating, for instance, mergers and acquisitions or bundling issues may limit the scope for a strategic use of asymmetries in knowledge and complementary assets. But, more interestingly (and somewhat provocative) even some usually-called market barriers can do this job. Theoretically speaking, all that is needed is a rule that directly or indirectly sets the boundaries of the business process and product containing the state-of-the-art knowledge. Within a second best context, rather than limiting the scope for innovative improvements some of these rules may act as a market coordination device able to deliver standardisation and knowledge di¤usion. The theoretical explanation proposed by the paper brings some standard developments of industrial organisation (IO) into a quality-ladders growth setting. Whilst strategic entry deterrence has been deeply analysed in IO works it has received much less attention in Schumpeterian growth models until recently. Some examples include explicitly defensive behaviour of incumbents such as the introduction of tacitness in the knowledge embodied in production techniques in order to prevent leapfrogging (Thoenig and Verdier, 2003) or the leader’s engagement in rent protecting activities such as patent blocking, intellectual property disputes and the like to delay their replacement (Dinopoulos and Syropoulos, 2007). In such papers …rms play simultaneously in a Nash-Cournot equilibrium and have symmetric technologies in R&D. These assumptions imply that the so-called Arrow replacement e¤ect holds and leaders do not continue to innovate because they take into account the loss due to their own replacement. There is, however, convincing evidence about the active rôle of leaders in R&D (see for instance Chandler, 1990; Malerba et al., 1997). In the proposed model, the participation of the leader in R&D contests arises endogenously. The circumstances under which the Arrow e¤ect vanishes has also been addressed in early in‡uent models of patent races (see for instance Gilbert and Newbery, 1982; Reinganum, 1983) but only recently spanned into quality-ladders literature. This de…nes a special class of models able to reproduce innovative leaders. This feature can be reproduced, for instance, thanks to the assumption of exogenous R&D advantages either within Nash-Cournot equilibriums and decreasing returns in R&D technology (Segerstrom and Zolnierek, 1999; Segerstrom, 2007) or within a Stackelberg type game (Etro, 2007 & 2008) that can be represented in a simple version with constant returns in R&D (Barro and Sala-i-Martin, 2004-Chapter 7). These works, however, consider innovation contests with exogenous asymmetries in R&D technologies. The explanation proposed here links these models with the above mentioned category by reproducing endogenous relative R&D advantages of leaders through defensive strategies. Introducing a stage in which R&D asymmetries can be acquired enriches the analysis of …rst-mover advantages. Similar strategic issues has been recently analysed by Grossman and Steger (2008). They show that, and competition: "Oceans Apart: Europe still seems to have less faith than America in the ability of the free market to tame monopolies".. 3.

(5) from the leader’s point of view, the erection of entry barriers and R&D are complementary activities. Despite entry blocking, this behaviour can be conductive to positive growth e¤ects when outsiders’ R&D do not generate knowledge spillovers. Important di¤erences exist between their model and the one presented in this paper: Cournot versus Bertrand competition here, deterministic innovation versus risky R&D investment here, a rather static entry barrier construction versus path dependency here (among others). This makes comparisons hard. Their results are, however, compatible with our equilibrium characterised by a permanent monopolist. Overall, what should be kept in mind is that, rather than render the analysis ambiguous, the richness of IO tools provides a highly selective level of robustness scrutiny. It is then not surprising to conclude that the relationship between product market regulation and innovation remains an open question. The rest of the paper is organised as follows. Section 2 discuss the empirical motivation and Section 3 the theoretical model. Finally, concluding remarks are presented in Section 4. In order to keep the exposition simple, most of technical developments is presented in appendix.. 2. An heterogenous e¤ect of product market regulation on R&D e¤ort. To empirically analyse the link between product market regulation and R&D investment, what is needed is to exploit the heterogeneity associated to di¤erent regulatory environments. The literature has then naturally relied on cross-country industry-level pseudo panel data and focused on aggregate outcomes. This is not a drawback per se as some phenomenons that are less visible at the individual level can be captured. Moreover, empirical exercises analysing the consequences of regulation at the industry-level face a lower risk of endogeneity since their focus is on de jure dimensions of competition rather than on outcome measures (e.g. concentration index or pro…t measures). This section examines a sample composed of 14 manufacturing industries across 14 OECD countries for the period 1987-2003. Figure 1 summarises the main …ndings that motivates the theoretical discussion that follows. It presents the correlation between R&D intensity (business R&D expenditure over value-added) and product market regulation. The latter is measured by the regulation impact indicator (REGIMP) constructed by the OECD that informs the extent to which industries are constrained by administrative burdens, entry regulation and other market barriers in key input sectors : network services; retail distribution and professional business services; and …nance. Both variables are available in time series for each country at the industry level (see Appendix A.2.1 for data description). The …gure presents added-variable plots obtained after an OLS regression that explains R&D intensity by regulation when country, industry and time …xed e¤ects are controlled for. The graphs shows the expected conditional residuals and, thereby, they presents a view of the relationship between regulation and R&D intensity which is free of any country-, industry- and time- speci…c R&D propensity. The coe¢ cients presented (the estimated slope) are then the same as in a "one-step" regression. The sample is split as follows. Panel A considers the 10% least regulated industries (in a given country); Panel B drops potential outliers of Panel A; Panel C considers the 25% most regulated industries; and Panel D the 25% most regulated industries that produce high-tech goods, de…ned as those industries using and producing ICT goods (29-34 ISIC-Rev.3). The pattern that emerges can be summarised as follows:. 4.

(6) 1. The e¤ ect of regulation depends on its level : in low-regulated environments (i.e. countryindustry couple) regulation is negatively associated to R&D e¤ ort (Panel A and B) while in highly-regulated ones it is positively associated (Panel C). 2. In highly-regulated environments, the positive e¤ ect of regulation on innovation is more important in activities using or producing high technology (Panel D).. Panel B (low-regulated & filtered). DNK-15t16 NLD-28 NLD-15t16 SWE-34 NLD-15t16 SWE-32NLD-29 NLD-15t16 SWE-32 NLD-28 SWE-34 DNK-15t16 SWE-32 NLD-28 SWE-32 NLD-28 SWE-28 SWE-34 SWE-34 UK-29 SWE-34 SWE-28 SWE-28 SWE-28 NLD-17t19 NLD-33 US-33 SWE-28 FIN-17t19 SWE-34 NLD-31 UK-29 SWE-32 NLD-33 SWE-34 US-21t22 SWE-30 US-33 SWE-32 FIN-17t19 DNK-25 US-21t22 FIN-17t19 SWE-34 NLD-17t19 FIN-25 US-32 UK-26 FIN-25 SWE-34 FIN-17t19 SWE-30 DNK-24 US-33 NLD-33 SWE-32 FIN-25 US-21t22 SWE-34 US-21t22 SWE-34 FIN-17t19 US-28 SWE-32 US-31 FIN-31 US-31 SWE-28 FIN-17t19 NLD-31 DNK-24 DNK-24 DNK-29 SWE-34 SWE-34 SWE-32 NLD-17t19 US-26 US-31 DNK-24 US-31 US-33 SWE-32 SWE-32 SWE-20 SWE-30 US-33 DNK-33 DNK-33 SWE-20 US-20 UK-26 FIN-32 DNK-29 US-32 DNK-17t19 FIN-32 NLD-17t19 DNK-30 SWE-30 DNK-30 US-31 SWE-32 FIN-31 UK-33 DNK-25 US-21t22 FIN-17t19 SWE-20 FIN-32 DNK-29 DNK-25 US-26 FIN-32 NLD-26 DNK-29 DNK-30 US-25 US-25 NLD-33 US-33 FIN-32 US-28 DNK-32 SWE-26 DNK-32 FIN-32 US-33 FIN-31 DNK-30 SWE-26 SWE-32 FIN-29 US-31 UK-33 US-32SWE-21t22 DNK-33 US-21t22 DNK-30 FIN-31 DNK-29 FIN-32 FIN-32 SWE-28 DNK-30 US-21t22 UK-26 DNK-33 FIN-32 US-26 DNK-31 UK-33 SWE-30 DNK-32 SWE-30 FIN-32 SWE-33 SWE-33 FIN-31 US-28 NLD-25 FIN-31 DNK-33 US-31 NLD-26 UK-28 DNK-26 US-33 UK-26 US-21t22 DNK-24 UK-28 SWE-28 SWE-21t22 SWE-21t22 US-33 US-31 FIN-17t19 US-25 US-32 SWE-24 US-31 SWE-21t22 US-21t22 US-25 SWE-17t19 US-33 SWE-30 US-21t22 US-25 FIN-30 FIN-30 DNK-32 US-26 US-25 US-25 SWE-17t19 US-20 US-31 SWE-25 FIN-31 US-15t16 NLD-33 US-33 FIN-32 SWE-30 NLD-25 DNK-33 SWE-28 US-32 SWE-30 US-31 SWE-33 SWE-24 US-32 US-32 NLD-26 FIN-32 SWE-29 NLD-24 SWE-29 SWE-25 SWE-33 UK-28 US-26 SWE-29 FIN-29 SWE-29 DNK-33 US-15t16 US-32 SWE-20 SWE-33 DNK-25 SWE-24 SWE-29 US-28 IRL-21t22 SWE-29 DNK-31 US-29 SWE-24 SWE-25 FIN-29 FIN-29 SWE-31 US-29 SWE-29 US-28 SWE-26 SWE-24 US-21t22 SWE-33 US-29 US-25 FIN-17t19 FIN-30 US-21t22 US-25 DNK-33 US-25 SWE-33 FIN-29 SWE-29 SWE-29 US-32 US-29 SWE-30 US-29 SWE-33 FIN-33 SWE-30 SWE-26 US-21t22 US-21t22 NLD-24 US-26 US-31 UK-28 US-25 DNK-25 SWE-25 SWE-29 SWE-33 US-31 US-15t16 US-32 SWE-17t19 DNK-30 SWE-24 SWE-17t19 DNK-33 US-15t16 FIN-33 SWE-17t19 SWE-30 US-32 US-32 UK-28 US-26 US-25 DNK-33 FIN-29 SWE-29 US-29 SWE-20 UK-17t19 US-25 US-31 US-25 UK-17t19 SWE-25 US-15t16 DNK-30 SWE-20 US-15t16 SWE-29 US-31 SWE-17t19 SWE-33 US-28 US-24 US-32 US-32 SWE-31 UK-17t19 US-29 US-29 US-21t22 FIN-29 SWE-30 SWE-17t19 FIN-30 FIN-33 SWE-30 US-24 SWE-33 UK-17t19 FIN-33 US-29 DNK-33 FIN-30 DNK-30 US-24 DNK-33 US-15t16 SWE-17t19 US-28 SWE-31 UK-25 US-28 US-28 FIN-33 US-24 SWE-30 NLD-21t22 SWE-15t16 SWE-15t16 SWE-17t19 FIN-33 US-24 DNK-32 FIN-33 SWE-15t16 SWE-17t19 SWE-15t16 US-15t16 NLD-21t22 US-29 SWE-33 SWE-17t19 US-28 SWE-31 SWE-26 SWE-31 US-29 US-29 DNK-21t22 US-24 SWE-26 US-29 UK-17t19 SWE-25 US-26 SWE-33 US-24 US-31 SWE-31 SWE-30 DNK-17t19 SWE-33 SWE-31 UK-25 US-28 US-29 US-29 FIN-34 US-20 SWE-31 SWE-31 SWE-20 SWE-15t16 FIN-30 SWE-25 FIN-30 SWE-17t19 SWE-31 SWE-25 SWE-25 SWE-31 US-17t19 FIN-34 FIN-34 UK-17t19 US-28 US-17t19 SWE-20 UK-25 FIN-33 SWE-26 FIN-34 DNK-26 SWE-15t16 US-17t19 NLD-21t22 FIN-34 SWE-26 SWE-15t16 SWE-31 US-28 US-28 FIN-34 US-28 FIN-34 UK-25 SWE-15t16 NLD-21t22 DNK-28 SWE-31 FIN-30 DNK-28 FIN-34 DNK-28 SWE-33 DNK-34 DNK-28 DNK-28DNK-21t22. e(log R&D intensity|X) 0 1. DNK-15t16. DNK-34 SWE-33 NLD-32 SWE-33. DNK-15t16 SWE-28 SWE-34 SWE-28 FIN-17t19 SWE-28 SWE-34 SWE-28 UK-29 SWE-28 NLD-28 SWE-34 US-21t22 FIN-17t19 SWE-34 SWE-34 SWE-30 US-33 FIN-25 FIN-17t19 SWE-32 NLD-15t16 US-21t22 FIN-17t19 UK-29 FIN-25 DNK-25 SWE-34 UK-26 DNK-24 NLD-15t16 NLD-15t16 FIN-17t19 FIN-25 US-21t22 FIN-17t19 US-21t22 SWE-30 US-33 SWE-34 US-28 SWE-34 NLD-28 FIN-31 SWE-32 US-31 SWE-28 US-33 NLD-28 SWE-32 DNK-24 NLD-29 DNK-24 NLD-28 DNK-24 US-26 SWE-32 SWE-34 SWE-34 SWE-34 SWE-32 US-31 US-31 DNK-29 FIN-17t19 SWE-30 UK-26 FIN-31 NLD-17t19 US-20 US-21t22 SWE-32 SWE-20 SWE-34 US-33 DNK-33 US-33 US-26 US-31 DNK-17t19 DNK-30 SWE-30 DNK-30 DNK-29 SWE-20 FIN-31 US-28 US-31 DNK-25 US-21t22 FIN-31 SWE-26 DNK-25 US-28 DNK-29 SWE-26 US-25 US-32 NLD-31 NLD-33 DNK-33 SWE-20 FIN-29 SWE-32 US-21t22 SWE-28 US-33 US-21t22 FIN-31 US-26 SWE-34 DNK-29 UK-33 NLD-33 DNK-30 DNK-30 DNK-30 US-33 FIN-31 FIN-17t19 DNK-29 FIN-31 FIN-31 NLD-17t19 US-21t22 DNK-24 SWE-32 UK-26 SWE-21t22 US-21t22 DNK-30 US-31 UK-28 DNK-26 SWE-28 US-25 SWE-21t22 SWE-21t22 US-31 UK-26 UK-28 US-25 FIN-30 US-25 DNK-31 UK-33 UK-33 SWE-21t22 DNK-33 FIN-31 US-26 FIN-30 US-31 SWE-24 US-33 SWE-17t19 SWE-17t19 SWE-32 SWE-30 DNK-33 US-31 US-15t16 DNK-33 US-31 US-33 FIN-32 US-25 FIN-32 US-25 SWE-32 US-32 SWE-33 US-31 US-33 SWE-33 FIN-29 NLD-33 SWE-25 UK-33 US-15t16 SWE-24 SWE-28 SWE-32 NLD-31 FIN-17t19 US-33 FIN-32 US-26 US-25 US-20 FIN-32 NLD-17t19 SWE-25 US-28 UK-28 NLD-31 SWE-32 US-21t22 SWE-30 SWE-33 DNK-25 DNK-33 FIN-29 SWE-30 US-28 SWE-26 US-21t22 FIN-32 DNK-33 SWE-29 SWE-24 SWE-24 SWE-30 US-25 SWE-29 SWE-24 FIN-32 SWE-29 FIN-29 FIN-32 SWE-20 SWE-25 FIN-29 FIN-29 SWE-29 US-29 DNK-31 SWE-29 US-25 IRL-21t22 FIN-32 SWE-29 SWE-29 SWE-29 DNK-32 SWE-30 US-28 DNK-33 US-31 FIN-30 SWE-29 US-26 US-25 US-21t22 US-29 SWE-33 SWE-32 FIN-32 FIN-32 SWE-33 NLD-17t19 DNK-33 US-31 SWE-26 SWE-24 US-33 US-29 US-25 US-29 US-25 US-15t16 SWE-33 DNK-32 SWE-17t19 US-29 SWE-25 SWE-32 US-26 DNK-25 US-15t16 FIN-33 SWE-31 NLD-26 UK-28 SWE-30 SWE-17t19 US-15t16 US-28 DNK-30 US-21t22 FIN-29 FIN-33 SWE-29US-32 SWE-30 SWE-33 SWE-17t19 US-25 US-24 US-15t16 US-21t22 US-31 DNK-32 US-29 UK-17t19 FIN-32 SWE-33 US-31 FIN-29 US-33 UK-17t19 UK-28 SWE-20 SWE-29 SWE-33 US-25 FIN-32 FIN-30 SWE-25 SWE-20 DNK-33 FIN-33 NLD-33 SWE-17t19 DNK-32 US-29 NLD-26 DNK-33 US-24 US-28 US-29 UK-17t19 US-24 DNK-30 SWE-31 SWE-17t19 SWE-29 SWE-30 US-32 US-32 UK-17t19 US-32 US-15t16 US-24 NLD-25 US-28 FIN-30 SWE-15t16 US-24 SWE-30 US-29 FIN-33 US-28 SWE-15t16 SWE-33 FIN-33 US-15t16 SWE-15t16 SWE-30 US-32 US-28 SWE-17t19 SWE-30 DNK-33 FIN-33 DNK-30 SWE-33 SWE-15t16 SWE-17t19 UK-25 FIN-33 NLD-26 FIN-33 NLD-24 SWE-31 SWE-17t19 NLD-25 SWE-31 US-32 SWE-17t19 US-29 US-29 NLD-25 US-29 US-32 US-24 SWE-26 US-24 SWE-26 US-32 SWE-33 DNK-21t22 US-26 NLD-33 SWE-31 SWE-25 US-28 US-32 US-29 UK-17t19 SWE-33 US-32 US-32 US-32 US-29 US-31 DNK-17t19 SWE-31 SWE-31 NLD-24 SWE-30 SWE-15t16 SWE-33 SWE-17t19 UK-25 FIN-30 US-20 SWE-31 SWE-20 FIN-30 FIN-34 SWE-31 SWE-31 SWE-25 US-28 SWE-25 SWE-31 US-17t19 US-17t19 SWE-25 UK-17t19 FIN-33 UK-25 SWE-26 FIN-34 DNK-32 SWE-20 FIN-34 SWE-15t16 FIN-34 US-29 NLD-21t22 DNK-26 US-17t19 NLD-21t22 SWE-26 FIN-34 SWE-15t16 SWE-31 FIN-34 US-28 US-28 US-28 SWE-15t16 UK-25 FIN-34 FIN-34 SWE-31 DNK-28 FIN-30 DNK-28 DNK-28 FIN-34 NLD-21t22 DNK-28 DNK-21t22 SWE-33 DNK-28 NLD-21t22 DNK-34. -3. e(log R&D intensity|X) -2 -1 0. 1. 2. Panel A (low-regulated). -4. -1. NLD-32 NLD-32 NLD-32. -.2. -.1 0 .1 .2 e(log Regulation |X). .3. -.2. coef = -1.1961441, se = .31468833, t = -3.8. -.1 0 .1 .2 e(log Regulation |X). .3. coef = -.80436196, se = .24171634, t = -3.33. 1. Panel D (high-regulated & high-tech). 2. Panel C (high-regulated). e(log -.5 R&D intensity|X) 0 .5. e(log R&D intensity|X) -1 0 1. BEL-21t22 BEL-21t22 ITA-34 ITA-34 ITA-34 BEL-21t22 NOR-17t19 ITA-30 ITA-34 ITA-30 ITA-30 ITA-34 ITA-34 ITA-34 ITA-34 ITA-34 NOR-17t19 ITA-34 ITA-30 ITA-30 ITA-34 ESP-20 NOR-17t19 ESP-20 NOR-17t19 NOR-17t19 ESP-21t22 ESP-21t22 NOR-33 NOR-17t19 BEL-21t22 ESP-21t22 NOR-33 ESP-21t22 NOR-33 NOR-17t19 ITA-24 NOR-26 ESP-21t22 NOR-30 ITA-24 NOR-32 ITA-30 NOR-29 NOR-29 ESP-20 NOR-26 NOR-30 NOR-17t19 ITA-21t22 ESP-21t22 FRA-26 NOR-32 ITA-24 BEL-26 NOR-26 BEL-26 BEL-26 ITA-28 BEL-26 ESP-20 ITA-31 ITA-29 ITA-31 ESP-21t22 FRA-26 FRA-26 NOR-15t16 ITA-25 FRA-26 ESP-20 NOR-15t16 ESP-21t22 NOR-15t16 ESP-28 NOR-30 NOR-25 NOR-29 NOR-32 ITA-25 ITA-24 ITA-28 ESP-28 NOR-26 ESP-29 NOR-26 NOR-25 FRA-34 NOR-32 ITA-31 NOR-26 ITA-21t22 BEL-26 NOR-29 FRA-26 ITA-32 ESP-25 ESP-29 ITA-25 ITA-30 ESP-28 ESP-20 BEL-24 ESP-25 ESP-28 ESP-30 ITA-25 ITA-24 ESP-26 FRA-26 FRA-34 ESP-21t22 NOR-29 NOR-32 NOR-33 ESP-15t16 ITA-25 ESP-28 NOR-32 FRA-26 BEL-24 ESP-25 NOR-31 NOR-24 ITA-32 NOR-31 ITA-25 FRA-26 FRA-34 FRA-34 NOR-24 NOR-15t16 ESP-24 NOR-29 ESP-28 ESP-26 ESP-31 ESP-30 ITA-31 ESP-31 FRA-26 ITA-28 ESP-15t16 ITA-24 ITA-24 FRA-34 ITA-25 ESP-26 NOR-33 ITA-29 ESP-30 ITA-25 NOR-25 ESP-26 BEL-24 ESP-34 BEL-26 FRA-34 ESP-28 NOR-25 FRA-24 NOR-25 NOR-33 ESP-26 ESP-25 NOR-15t16 FRA-34 ESP-20 NOR-15t16 ESP-28 ITA-21t22 ITA-24 ITA-24 FRA-34 ESP-29 FRA-24 ESP-15t16 ITA-24 ESP-25 ESP-25 ESP-31 ESP-28 ESP-26 FRA-26 ESP-28 FRA-24 ESP-34 FRA-26 FRA-34 ESP-25 NOR-33 ESP-24 NOR-24 ESP-30 NOR-33 UK-30 IRL-26 NOR-15t16 ITA-24 ITA-31 ITA-25 NOR-30 NOR-30 FRA-24 ESP-34 ITA-31 ESP-15t16 BEL-21t22 ESP-26 ESP-29 ESP-31 NOR-24 ESP-15t16 ESP-24 NOR-24 ESP-24 ITA-29 ESP-28 ESP-26 ESP-26 ESP-34 NOR-33 NOR-24 ITA-29 FRA-24 ITA-15t16 ESP-24 ESP-31 ESP-24 ESP-29 NOR-32 ESP-24 ESP-15t16 NOR-31 ESP-20 FRA-21t22 ESP-26 ESP-26 ESP-29 NOR-24 ESP-24 BEL-21t22 NOR-30 FRA-21t22 ITA-28 ESP-29 ESP-20 NOR-33 ESP-15t16 ESP-15t16 ESP-24 FRA-21t22 ESP-24 ESP-30 ESP-32 FRA-21t22 ITA-15t16 NOR-34 NOR-33 FRA-21t22 ESP-20 ESP-26 ITA-21t22 ESP-17t19 NOR-25 ITA-30 ITA-28 NOR-31 NOR-33 NOR-30 ITA-31 FRA-21t22 ESP-30 ESP-34 ESP-24 ITA-15t16 ESP-32 FRA-21t22 FRA-21t22 ITA-29 ITA-15t16 ESP-31 ESP-34 ITA-29 FRA-21t22 NOR-34 NOR-32 ITA-29 ESP-15t16 NOR-25 NOR-32 ITA-29 ITA-31 FRA-21t22 ITA-28 NOR-24 NOR-32 NOR-30 FRA-21t22 ESP-32 ESP-34 ESP-17t19 NOR-30 NOR-30 NOR-32 ESP-24 ITA-30 NOR-24 ITA-30 NOR-24 NOR-33 NOR-30 NOR-30 NOR-24 NOR-33 ITA-17t19 NOR-34 ITA-26 ESP-20 ITA-26 NOR-25 ESP-31 ITA-21t22 ESP-17t19 ITA-30 ITA-21t22 NOR-34 FRA-29 NOR-30 NOR-25 NOR-33 ESP-34 ITA-26 FRA-29 NOR-34 ITA-28 ESP-30 NOR-30 ITA-26 NOR-25 ITA-28 NOR-25 ITA-26 NOR-34 NOR-30 ITA-26 ITA-26 ITA-26 NOR-34 ITA-26. -2. ITA-17t19BEL-34 ITA-17t19BEL-34 BEL-34 BEL-34 ITA-17t19 BEL-34 ITA-21t22. -3. -1. ESP-20 ITA-21t22 ITA-21t22. -.15. -.1. -.05 0 .05 e(log Regulation |X). .1. ITA-34 ITA-34 ITA-30 NOR-29 ITA-34 ITA-34 ESP-29 NOR-32 ITA-34 NOR-30 ESP-29 ITA-30 ITA-34 NOR-29 ITA-30 NOR-32 ITA-30 NOR-29 NOR-30 NOR-33 ITA-34 ITA-30 NOR-30 ITA-34 ITA-34 NOR-29 FRA-34 NOR-33 ITA-34 NOR-29 NOR-31 NOR-32 NOR-33 NOR-33 ESP-30 FRA-34 ESP-31 NOR-30 ESP-31 NOR-29 BEL-34 ESP-30 NOR-33 NOR-32 ESP-30 ESP-31 FRA-34 NOR-31 ITA-29 ESP-31 BEL-34 ESP-29 NOR-33 FRA-34 ITA-30 ESP-31 ESP-29 NOR-32 NOR-30 ESP-34 NOR-33 FRA-34 ESP-29 NOR-30 FRA-34 NOR-30 ESP-29 NOR-32 ESP-29 ESP-31 NOR-31 NOR-33 NOR-32 ESP-30 NOR-32 NOR-33 UK-30 ESP-34 NOR-32 ITA-30 NOR-33 ESP-34 NOR-30 NOR-31 ITA-29 BEL-34 FRA-34 ITA-31 NOR-30 NOR-32 FRA-34 FRA-34 ITA-31 NOR-33 NOR-32 NOR-33 FRA-34 ESP-34 ITA-31 ITA-31 ESP-34 NOR-30 NOR-30 ESP-30 ESP-32 ITA-31 ESP-34 BEL-34 ESP-31 BEL-34 ITA-31 ESP-32 ITA-31 ESP-32 NOR-30 NOR-30 NOR-34 ITA-32 ESP-34 ITA-31 NOR-30 ESP-34 FRA-29 ITA-32 NOR-34 ITA-29 ITA-29 ESP-30 ITA-29 FRA-29 NOR-33 NOR-30 NOR-33 ITA-29 ITA-29 ITA-30 NOR-34 ITA-30 ITA-30 ESP-34 NOR-34 NOR-33 NOR-34 ITA-29 ITA-30 NOR-34 NOR-34. -.1. coef = 3.4444043, se = .75437453, t = 4.57. -.05 0 .05 e(log Regulation |X). .1. coef = 6.2681456, se = 1.0931101, t = 5.73. Figure 1. Regulation and R&D intensity In Appendix A.3 this evidence is completed with (i) several tests on the di¤erentiated impact of regulation on the innovative activity of high-tech industries, namely heteroskedasticity, endogeneity issues and …xed-e¤ect vector decomposition to analyse a time-invarying variable that identi…es highttech industries; and (ii) interquantile regressions to test the di¤erence in …ts at di¤erent quantiles of the conditional distribution.3 Overall, the pattern depicted by Figure 1 remains robust to this variety of robustness tests. Indeed, the scope for innovation-boosting e¤ects of regulation has been well documented in the empirical 3. In a previous working paper version (Ledezma, 2009), similar positive correlations between regulation and R&D intensity in high-tech industries are obtained using a narrower de…nition of high-tech goods, alternative indicators of regulation and several time-varying explanatory variables (…nancial deepness, capital intensity, R&D spillovers and the proximity to the world technology frontier). 5.

(7) literature. Namely, the widely-admited idea about a negative impact of product market regulation on innovation and productivity, which is supposed to be specially pronounced in leading-edge industries, is less widely supported by empirical studies. Using the OECD indicators of economy-wide product market regulation, Nicoletti and Scarpetta (2003) report a positive interaction between regulation and the proximity to the world technology frontier that signi…cantly explain multifactor productivity growth. The authors interpret their …nding as a negative e¤ect of market barriers on the catchingup process. However, their results also implies that the impact of regulation on productivity growth positively increases with the proximity to the technology frontier. Using comparable proxies, in Amable et al. (2010) we …nd no evidence of a negative e¤ect of regulation on patenting close to the world technology frontier. After several robustness checks, what remains is that the marginal e¤ect of regulation on innovation tends to be positive at the leading edge.4 Inklaar et al. (2007) analyse several sources of multifactor productivity growth in service sectors. Excepting telecommunications, their results fail to show a statistically signi…cant negative correlation between OECD market barriers indicators and productivity growth. Arnold et al. (2008) combines industry-level regulation data with …rm-level information. They report that regulation induce a negative e¤ect on …rm productivity for …rms above the median productivity level observed on the national distribution. This di¤erent way to de…ne the closeness to the technology frontier may explain the di¤erence with the previously mentioned results. Gri¢ th et al. (2006) investigate the e¤ect of the Single Market Programme (SMP) on R&D expenditure. The authors construct indirect indicators of market regulation through a step function that, basically speaking, seeks to measure the expected exposition to the SMP. Pro…tability e¤ects of regulation are isolated following a two-step methodology. Overall, results are that the liberalisation trend induced by the SMP is positively correlated with R&D investment. However, in several of R&D reduced-form regressions (i.e. those including other channels than pro…tability), the group of industries related to the category of "high-tech public procurement" (including telecommunications equipment, o¢ ce machinery and medical and surgical equipment) presents a signi…cantly negative correlation, that is to say, a negative impact of deregulation on R&D and productivity (as in panel D of Figure 1 ). Results presented in Figure 1 adds to this global picture two important ideas concerning the ambiguity of the e¤ect of regulation on innovative e¤ort. Firstly, this link not only depends on relative performances (i.e. the distance to the technology frontier) but also on the technological nature of manufacturing activities. Secondly, a di¤erentiated e¤ect of regulation arises depending on the level of regulation itself: regulation may exert positive in‡uences on innovation incentives, but only after attaining a certain level.5 4 These industry-level works contradict micro-level results found by Aghion et al. (2005) for a panel of UK …rms using pro…tability-based measures of competition. The inverted-U shape pattern between competition and innovation, that underlies Aghion et al.’s (2005) claim, has also been empirically relativised at the micro-level by Tingvall and Poldahl (2006). A number of theoretical arguments, dealing with strategic behaviour, can be mentioned to explain the lack of clear-cut results on this matter (see for instance Etro 2007, Chapter 4; Tishler and Milstrein, 2009; Amable et al. 2010) 5 This …nding is consistent with macro-institutional literature emphasising the diversity of capitalism (Hall and Soskice, 2001; Amable, 2003). Di¤erent institutional con…gurations are able to deliver economic performance. In some of them, market-based forces ensured by a deregulated environment are the key dynamic engine, in other it is rather institutional coordination that is associated to economic performance.. 6.

(8) 3. The model. The formal setting uses a semi-endogenous quality-ladders model without scale e¤ects. The basic setup is based on Li (2003) which generalises Segerstrom’s (1998) framework. It considers imperfect interindustry substitutability and remove steady state scale e¤ects by assuming that, as quality improves, new discoveries need more R&D e¤ort. At equilibrium the innovation rate will not depend on the size of labour allocated to R&D but on the rate of population growth.6 As it will be clear in a moment, this semi-endogenous steady-state innovation rate underlies the one that would be obtained in the absence of strategic R&D barriers. The section is structured as follows. Section 3.1 begins with the basic setup of consumption and production. The core of the model is then presented : the strategic setting in Sections 3.2 and 3.3 and the e¤ects on aggregate R&D e¤ort at equilibrium in Section 3.4.. 3.1. Consumption and production. The economy is composed of identical dynastic families whose number of members grows at the exogenous rate n > 0. Each member of a household supplies inelastically one unit of labour. Without loss of generality, initial population is set to 1, so that the population at time t is L(t) = ent . Consumption decisions can be separated into instantaneous and intertemporal programs. Instantaneous decisions:. P. Per capita utility at each time t is given by the CES formulation: 2 1 Z 4 u (t) = z (t; !) 0. 1. 3. d! 5. 1. (1). j d (j; t; !) is the sub-utility function associated to each industry !. The demand for z (t; !) j the good of quality j at time t in industry ! is denoted by d (j; t; !). The term j captures the quality level j of a given good, where > 1 is a parameter representing the size of quality upgrade. Thus, within a given industry consumers preferences are ordered by the quality of the available varieties. To avoid confusions in notation, all round brackets, () ; are reserved to the arguments of the functions of the model. At any time, individuals allocate their consumption expenditure E (t) seeking to maximise u (t). This static problem can be separated in two components: a within-industry consumption decision and a between-industry one. Giving the utility function z (t; !) for the quality varieties in each industry !, all intra-industry expenditure will focus on the good j having the lowest quality-adjusted price: p( j ;t;! ) j = arg min : j (j). The between-industry problem concerns the allocation of total expenditure E (t) among all ! 2 [0; 1]. This consists of applying the optimal intra-industry demand z (t; !) = d (j ; t; !) to (1) and 6. This feature characterises a second wave of quality-ladders models that solve problems of scale e¤ects in the steady state growth (Segerstrom, 1998; Young, 1998), a property strongly contradicting empirical evidence found by Jones (1995): while resources allocated to R&D increase exponentially in the long-run data, productivity growth remains almost constant. For a survey on the evolution of this type of schumpeterian models see Dinopoulos and Sener (2007).. 7.

(9) maximising u (t) subject to. Z1. p (j ; t; !) d (j ; t; !) d! = E(t), which leads to the well-known CES. 0. demands: (j ; t; !). d (j ; t; !) = p (j ; t; !). R1. (j. ; t; ! 0 ) p (j. E(t) ; t; ! 0 )1. (2). d! 0. 0. Where (j ; t; !). j [. Intertemporal decisions:. 1]. is a quality-level index.. Using a subjective discount rate. > n; each dynastic family maximises. its intertemporal utility. U. Z1 = e. [. n]t. log u (t) dt. (3). 0. subject to a (t) = w (t) + r (t) a (t). E (t). na (t). where a (t) is the endowment of per capita assets. Its variation a (t) is decomposed into current wage income of the representative household member w (t) plus stock market gains r (t) a (t) minus expenditure E (t). Between t and dt, the growth of per capita assets needs to be adjusted by population growth n. Solving this program leads to the following optimal intertemporal rule: E (t) = r (t) E (t) 3.1.1. (4). Producers and price setting. Labour is the only factor in production and is used in a technology with constant returns to scale. Each …rm producing the variety ! sells its output to all members of the representative household. Thus, the …rm produces a quantity of d (j ; t; !) L (t) ; sells at price p (j ; t; !) and incurs a production cost w (t) d (j ; t; !) L (t). After wage normalisation, w (t) = 1; the pro…t of each producer is given by: (j ; t; !) = [p (j ; t; !). 1] d (j ; t; !) L (t). (5). Standard monopolist pro…t maximisation would lead to a markup over marginal costs: p (j ; t; !) = 1 . However, the monopolist is also in competition with …rms o¤ering lower quality goods. Bertrand competition yields to a limit pricing behaviour. Consider, namely, a …rm laying one step down the = j 1 1 (i.e. its price leader in the quality-ladder and whose best quality-adjusted price is p(j j 1;!;t) 1 equals its marginal cost). The leader …rm will then charge p (j ; !; t) = and get all demands. A tie-break rule assumption, stating that a consumer facing similar quality-adjusted prices prefers the good with the highest quality, allows to avoid the use of a quality-adjusted price in…nitesimally lower. 8.

(10) The application of this intra-industry price setting will depend on the size of innovation and the monopolist power …rms will charge p (j ; !; t) = . On the contrary, if the 1 . If 1 > 1 leader is unconstrained to charge its optimal monopolistic price rule p (j ; t; !) = 1 . This introduces the following assumptions:. Assumption 1 p= :. Price setting is constrained by outside competition. 1. >. , so that p (j ; !; t) =. Assumption 2 Knowledge spillovers are such that any time an innovative …rm succeeds, the previous version of the good is available for the rest of …rms. The model works with quality upgrades of the same good. In this sense, it is more plausible to suppose, as in Assumption 1, that the size of each upgrade is not big enough to induce the innovator to adopt the same price behaviour than a monopolist having no outside competition. Assumption 2 implies that there always be a …rm one step down so that the only possible price setting is this bounded limit price. Therefore, an asymmetry in price setting incitations will not arise here. Denicolò (2001) shows that when innovation is non-radical and the leader …rm can sustainably obtain a gap of at least two steps, it can practice a monopolist price while the next competitive outsider engage in Bertrand competition with consequently lower incitations. Here, on the contrary, knowledge spillovers constrain the sustainable technology gap between …rms to be one step. By assuming this, we focus on endogenous R&D advantages as the source of innovation incentives. Even if small, these R&D cost advantages justify that the leader continuously invests in R&D (Klette and Griliches, 2000). Putting demands (2) into leader pro…ts (5) and using the fact that p neither depends on j nor on ! yields: (j ; !; t) = Where Q (t). R1 0. (j ; !; t) d! =. R1. j [. [p. 1] (j ; !; t) E (t) L (t) p Q (t). 1] d!. (6). is the average quality index. Thus, the monopolistic. 0. competition framework implies that …rms compete in quality with the whole economy.. 3.2. R&D technologies and quality improvements. At each state-of-the-art quality level j; the successful innovator of the current R&D race improves quality to the level j + 1 and climbs the quality-ladder one step up.7 The above-exposed price setting implies that the successful innovator becomes the sole producer in the industry. Thus, each incumbent is also the monopolist and the leader of the industry. Di¤erently from the standard setup, in this model the incumbent does not wait until the next innovator "steals" its rents, but seeks to deter its potential rivals and to remain in the market. This subsection is devoted to set the underlying R&D framework allowing for these mechanisms. Before starting a subscript simpli…cation can be made. 7. More generally, it is usually supposed that, at t = 0, the state-of-the-art quality in each industry is j = 0 and that some producer has the knowledge to fabricate a good of quality j = 0: Firms then engage in R&D races to discover a new version of the good.. 9.

(11) Remark 1 Observe that (i) there is only one …rm producing a positive quantity in an industry; (ii) the only di¤ erence among industries concerning state variables is the current state-of-the-art quality level j; and that (iii) all endogenous variables (except prices) depend on t . Based on (i) and (ii), j! will now summarise the couple (j ; !) ; which indicates the current stateof-the-art good produced by the leader of industry !. Thanks to (iii) the time index can be dropped, keeping in mind the time dependency of the model. 3.2.1. Quality dimensions. Quality is represented as a multidimensional state variable. The aim is to model in a simple way the presence of defensive strategies available in the process of innovation itself. This is not possible in the standard quality-ladder setting where innovations are technologically neutral as each innovator climbs the quality ladder in the sole possible manner. Here, it can chose a non-neutral direction to do so. As it shall be clear in a moment, this multidimensional representation will be heavily reduced to a scalar measure. The quality provided by a …rm producing in industry ! is given by the quality vector q (j! ) = fq1 (j! ) ; q2 (j! ) :::; qm (j! )g . These m dimensions may concern not only the fabricated good but also the whole business process involved in the provision of the good to the customer ands related services. m P The level of quality is summarised by the euclidean norm of the vector kq (j! )k = qk2 (j! ) and k=1. the quality mix by its direction (j! ), the angle of the vector. The latter captures the composition of the o¤ered good. In line with the intra-industry sub-utility function, di¤erent mix concerning the same industry and quality level are also perfect substitutable versions of the same product. In each industry, two di¤erent quality mix provide the same utility if their magnitude is equal. Direction only matters in the research sector. The quality state j! is the outcome of step-by-step innovations. The magnitude of the quality vector is upgraded at each step by a factor of ; the size of innovations. The quality provided by the state-of-the-art j! is thus de…ned as kq (j! )k = j! : 3.2.2. R&D technologies. The de…nition of R&D technologies is based on two assumptions: Assumption 3.. Only the current successful innovator can change the quality mix.. Assumption 4. Outsiders take a time to acquire the knowledge and capabilities embodied in the new dimensions of quality featured by the new state-of-the-art good. Assumption 3 re‡ects the innovator’s advantages arising from its private knowledge about the new product. Once the new discovery come o¤, the new blueprint is certainly known by the innovator. The leader …rm now has the choice about what visible properties its product and related services will have in the market. Assumption 4 seeks to represent the fact that challengers are compelled to provide a new business solution in the context of several disadvantages concerning learning, experience, lead time 10.

(12) developing, lack of codi…cation, etc. (for short knowledge ) as well as unfavourable conditions related to the need of new manufacturing complementarities, patents and licenses, agency and organisational issues, market access, etc.(for short capabilities). They need to overcome these barriers in order to be able to replicate and to improve the current state-of-art good. Outsiders’R&D technology Outsiders carry out R&D activities by using labour as input. R&D is governed by a Poisson stochastic process: ì units of labour allocated to research during an interval of time dt imply a probability of success 0 (j! + 1) ì dt of a new upgrade. We call R&D productivity 0 (j! + 1) the augmenting factor of the instantaneous probability of success of one R&D unit of labour. For the outsider, this R&D productivity is de…ned as 0 (j!. + 1). h cos j! (j! + 1). Following Li (2003), the R&D productivity is a function of the upgrade endeavoured. The presence of the quality index (j! + 1) = [j! +1][ 1] represents the idea that, as the level of quality increases, the next improvement becomes harder and R&D more costly. h is an exogenous technological parameter of R&D e¢ ciency. What is di¤erent here is the incidence of a change in the quality mix. This change is captured by the normalised scalar product between the current and the previous quality vector: q (j! ) q (j! kq (j! )k kq (j!. 1) = cos 1)k. j!. where j! = (j! ) (j! 1) is the angle between q (j! ) and q (j! 1). Recall that the cos ( ) function is symmetric and monotonically decreases from 1 to 0 along with j j! j 2 [0; =2[ (in radians). Hence, introducing a new good with a quality mix j! far away from the previous one, reduces the outsiders’probability of R&D success by a factor cos j! < 1 ; where > 0 captures the impact of this technological bias. The instantaneous probability of innovation Ii implied by the R&D e¤ort of outsider i is then: Ii = ì. h cos j! (j! + 1). (7). Remark 2 Between two waves of innovation, represented by two vectors of quality, the speci…cation of quality dimensions is not needed. The e¤ ect of technological bias collapses to the scalar product among both vectors. All information is contained in the angle between them. Leader’s R&D technology The leader …rm does not face the di¢ culty coming from bias. It has discovered the current state-of-the-art product and it is the sole producer that knows how to incorporate the new dimension in the manufacturing of the good. Hence, the leader’s R&D productivity has the standard functional form: L (j!. + 1) =. 11. h (j! + 1).

(13) 3.2.3. Regulation and the path of innovation. As a way to protect their position, the innovator can add a new quality dimension to the current mix. The model assumes that the discovery enables the innovator to incorporate some new component or process that exploit its asymmetries in knowledge and capabilities. This new dimension can include, for instance, new services, bundling, speci…c intermediate inputs, manufacturing installations, property rights, key suppliers, vertical integrations, etc. introduced with the aim to take advantage from their complementary assets, know-how and technology. Quality is of course upgraded by innovation, but over a speci…c path, strategically chosen.8 Figure 2 illustrates the path of innovation. Let us start from the quality level j in a given industry. At this stage thee good is totally based on dimension q1 (implying a horizontal vector). Once the next innovative …rm has succeeded in upgrading the quality level to j + 1; it introduces a bias by including dimension q2 : The …rm then produces the new version of the product with a quality vector having a direction j+1 far away from the previous one. By doing so, it increases the di¢ culty of the next R&D race (the one leading to the j + 2 level) by a factor of cos j+1 : Then, the next innovation occurs and improves the quality level to j + 2 and do the same: Since at each discovery new dimensions are available, this process may last inde…nitely. The …gure suggests a case of "re-discovery" of quality dimensions. Dimension q1 has been dropped in the j + 2 wave (q1 (j + 2) = 0) and re-introduced again by the winner of the j + 3th contest. 9. q3 q2. j+2 θj+2 j+1 j. j+2 θj+1. q1. j+1 Figure 2. Innovation Path. 8. For instance, Intel Inside has recently incorporated the hafnium, a new material allowing to concentrate more transistors into their microchips (45 nm processors) . This requires investments in manufacturing adaptations that give the upper-hand of Intel over its rivals. 9 This possibility of rediscovery (i.e. the incorporation of old dimensions into the current state-of-the art- product) can conceptually substitute a continuous process of discovery of quality dimensions. Hence we do not need to go further in the speci…cation of the R&D process, as long as the way to include old characteristics in the new version is by no means immediately feasible. Think for instance in compatibility issues involved in the inclusions of Dolby technology in i-Pods or in licencing-out market imperfections issues, accounted by Zuniga and Guellec (2009)’s survey analysis.. 12.

(14) Any leader that changes the mix incurs a variable cost (in units of labour) of adapting the new version. This cost is de…ned through the following functional form : c(. j! ;. ). f cos. L (j! ). j!. (8). where > summarises the extent to which product market regulation limits the new version of the product by increasing the cost of changes in the quality mix. Hence, product market regulation is seen as a device constraining the way in which the new discovery is introduced into the market.10 The functional form in (8) imposes that regulation rises the marginal cost of bias. It acts as a decreasing monotonic transformation of the e¤ect of technological bias on outsiders’R&D productivity. Therefore, the assumption > means that regulation is supposed to be e¤ective in limiting the new incumbent. Cost function (8) also assumes that the cost of introducing a technological bias in the new product diminishes with the R&D productivity involved in its discovery. This is captured by the term L (j! ) : the R&D productivity of the leader …rm in the former R&D race j! (the one that it has won). The presence of L (j! ) also implies that high-quality goods are more di¢ cult "to bias" since R&D productivity decreases with the quality level of the industry. Finally, a cost parameter f is included to take into account the measure of units of labour required to activities relating to defensive strategies.. 3.3 3.3.1. Strategic behaviour Stochastic jumps and timing of the Stackelberg game. If researchers of an outsider …rm i succeed, the …rm becomes the leader and gets a value denoted by vL (j! + 1) : Free entry in the research sector implies that …rms enter up to the point where the expected value of innovation, vL (j! + 1) o (j! + 1) ìo dt; equates the R&D e¤ort ìo invested during the in…nitesimal interval of time dt. That is to say : vL (j! + 1) =. 1 o (j!. + 1). (9). Given the CRS in R&D technology, the R&D e¤ort of the outsider for a given value of a successful innovation vL (j! + 1) is then: 8 if vL (j! + 1) < o (j1! +1) > < 0 1 if vL (j! + 1) > o (j1! +1) ìo = (10) > 1 : ` 2 R+ if v (j + 1) = io L ! o (j! +1) P Let `0 = i ì0 be the total amount of R&D carried out by outsiders. The Bellman equation of a (potential) innovative leader can be written as 10. For instance, some of the practices captured by the indicator REGIMP (see section 2) in retail and professional services include limitations such as licensing permits or restrictions on the type of products that can be o¤ered. Simillarly, practices in utility sectors leading to regulated entry and almost inexistent import penetration will probably be a de facto limited room for technological bias.. 13.

(15) rvL (j! ) =. L. `L + IL [vL (j! + 1). vL (j! )]. Io vL (j! ). c(. j! ;. ). (11). If the leader invests `L in R&D, with instantaneous probability IL = `L L (j! + 1) its optimal value vL (j! ) can jump to vL (j! + 1) thanks to the new discovery. With instantaneous probability Io = ò o (j! + 1) the leader may be replaced by a successful outsider. In the meantime, the leader …rm enjoys its monopolist pro…ts L and pays `L unit of labour for new discoveries as well as c ( j! ; ) units of labour for defensive strategies. Equation (11) implicitly says that the current value of a follower is zero. This is the result of Bertrand competition, the free entry condition with CRS and zero R&D sunk cost to be paid before playing. Built on this setting, the timing of the model is as follows. At the very beginning of the technological state j! , we have the leader (the current successful innovator), symmetric R&D technologies and the parameters of the model, namely the level of regulation. Free entry in the research sector applies. Then the following stages take place: 1. Entry into the product market: the leader chooses the level of bias j! in order to maximise its value. It takes the parameters of the model as given, namely the level of product market regulation. Production starts. 2. R&D race: (a) The leader decides its optimal level of R&D e¤ort `L taking outsiders reaction (10). j!. as …xed and knowing. (b) Having observed the leader’s commitment `L , outsiders set their optimal R&D e¤ort ò Once investments are engaged, they remain …xed during the contest. Namely, the leader cannot change its choice of j! . The ‡ow cost c ( j! ; ) can then be seen as the amortised defensive investment per-time interval dt. Stage 3 is the core of the Stackelberg game based on Barro and Sala-i-Martin (2004) model. A key di¤erence is that here the relative cost advantages are endogenous thanks to stage 2. 3.3.2. The R&D race. The following proposition establishes the conditions under which outsiders are deterred. The presentation focuses on the case where j! = is constant. In section 2.3.3, this will prove to be true for a constant outsider menace, which is the standard steady state condition of this kind of model.. Proposition 1 For a constant value of R&D e¤ ort for outsiders is cos. j!. = ; a su¢ cient condition to ensure a non pro…table h. 1. 14. [. 1]. i. (12).

(16) Under this condition, the leader’s R&D e¤ ort can be positive and irrespective of outsider actions. In this equilibrium the leader value and the interest rate verify, respectively vL (j! ) =. r=. L. c( ; ) r [. 1E L 1. p p. 1]. (13). h. (14). Q. Proof. See Appendix A.1.1. The inequality stated in (12) will be referred to as the credibility condition. Intuitively, it de…nes L (j! +1) a threshold for the R&D productivity advantage of the leader = cos1 , which is increasing o (j! +1) in (i.e. decreasing in cos ). This threshold determines whether the R&D investment is pro…table for the leader …rm (i.e. whether it is credible). If this is the case, constant returns of R&D investment imply that the leader can potentially perform enough R&D e¤ort to put outsiders out of competition. Thus, when the bias is strong enough, the leader does carries out research e¤ort and the outcome is that the value of the next quality improvement is lower than the R&D cost incurred by outsiders vL (j! + 1) < o (j1! +1) (see the proof of Proposition 1 for details). As a consequence, outsiders react by setting zero R&D e¤ort, meaning no replacement menace: Io = 0. In this case the leader value is given by (13). In contrast, if the credibility condition does not hold, the leader will be replaced and all R&D will be done by outsiders. Its value in such a situation is that implied by (11) for `L = 0: The ex ante value of the incumbent can be summed up as:. vL (j! ) =. 3.3.3. 8 > < > :. L. c( ; ) r+Io. if cos. [. > 1. 1]. (a) (15). L. c( ; ) r. if cos. 1. [. 1]. (b). Entry into the product market. At the moment in which the leader enters into the product market (i.e. when it introduces the new good), outsiders can potentially carry out research e¤orts and the free entry condition in the research sector holds. Thus, the rationale of the decision of bias starts by considering that, at this stage, no technological advantage has been acquired. The leader …rm is not credible for the moment and its value is given by (15,a). Potentially, a new successful innovator can replace it. But the leader can make this task harder. A higher R&D di¢ culty means a lower probability of replacement and then a higher expected value. This decision of bias implies a cost of c ( j! ; ) units of labour which is increasing in , the regulation parameter. The leader …rm will choose a value of j! that maximises its value. By the maximum principle, …rst order condition for j! on the RHS of the leader’s Bellman equation (11) leads to @Io vL (j! ; @ j!. j! 1 ). 15. =. @c ( j! ; ) @ j!. (16).

(17) Hence, the marginal reduction of the expected loss of the leader must equal the marginal cost incurred in defensive strategies aimed at obtaining that reduction. Note that j! 1 is explicitly indicated as an argument of vL . If free entry holds, then vL should depend on the investment required in the previous step and so on j! 1 . Equation (16) implies then a dynamic sequence relating j! to j! 1 : It is shown in Appendix A.1.2 that this sequence converges. It is clear from (16) that the emergence of this path dependency does not rely on functional form assumptions. On the contrary, the assumed forms of R&D productivity and cost function help to obtain a closed solution. A key element is the presence of L (j! ) in equation (8) whereby high-quality goods are more di¢ cult to bias. This assumption allows to o¤set the e¤ect of the level of innovation that is also present in vL (j! ). The following proposition states the main properties of the stationary level of bias. Proposition 2 De…ne IoL ò L (j! + 1) as the "potential menace of outsiders", that is to say, the probability of outsiders’innovative success in the absence of bias: When free entry in the research sector holds (i.e. the leader is not credible), there exists an optimal choice of bias if its impact ( ) is high enough. Its value is constant for a constant potential outsider menace (IoL ) and it is given by 1. f. cos =. IoL. (17). Proof. See Appendix A.1.2. As expected cos decreases with IoL : A higher potential menace of replacement implies a more aggressive defensive strategy. Moreover, for a given value of IoL one veri…es @ @cos > 0: As expected, regulation reduces the bias. Recalling that outsiders’ probability of R&D success is Io = IoL cos ; we obtain: Io = IoL. f. (18). This hazard rate converges toward its potential, Io ! IoL ; when ! 1. This means that a high level of regulation may (asymptotically) eliminate the bias (cos ! 1). In particular, can determine whether the credibility condition holds. For a low enough level of regulation, the bias delivers credibility. In that case the economy jumps to a permanent monopolist framework with an innovative leader whose value is that of equation (15,b). The choice of in this limit case is established by Proposition 3.. Proposition 3 When the leader credibility is ensured, the optimal value of bias is h i [ 1] cos = 1 Proof. It follows immediately from (15,b).. 16. (19).

(18) Here the incumbent enjoys permanent pro…ts as an innovative monopolist. Once this dominant position has been achieved, higher level of bias will only increase costs without additional value. Therefore, the leader does not need further R&D advantages beyond the credibility point. Thus, two type of equilibriums can arise. In the …rst case, outsiders do all R&D and the leader delays its replacement in a Schumpeterian replacement equilibrium (SR). In the second situation, the leader may become the sole innovator enjoying permanent pro…ts in a Permanent monopolistic equilibrium (P M ). To identify the underlying conditions of each type of equilibrium, the schedule of decisions studied so far needs to be completed with the macro steady-state analysis. This is what the next section does.. 3.4. Global accounting and steady state. For a constant level of bias, the dynamic of the model is that of a standard semi-endogenous growth setup and has a symmetric steady-state equilibrium. Starting from the innovation rate observed at this equilibrium, this section derive the underlying potential menace of outsiders (IoL ) and the consequent defensive reaction of the leader. Having found this, we can analyse how regulation a¤ects the steady-state choice of the leader and, thereby, the steady-state equilibrium itself. 3.4.1. The Schumpeterian replacement equilibrium. The macro equilibrium for a continuum Schumpeterian replacement is given by the ful…llment of the labour market clearing and the free entry condition in the research sector. Labour market clearing R1 under full employment needs the addition of labour used in research Lr = ò (j! + 1) d!, manufac0. R1. R1 turing Ly = L d (j! ) d! and defensive activities related to technological bias Lf = c ( 0. ;. ) d!. The. 0. focus here is the symmetric steady state equilibrium in which expenditure E and outsiders innovation rate I0 are constant. The latter implies that I0L is also constant and the same across industries. The same is then true for cos . Using the de…nition of the average quality index Q introduced in equation (6) and the quality index of each industry (j! + 1) = [j! +1][ 1] , the demand for labour in research activities is given by: 1 Io Q h cos After including demand equation (2), labour required for manufacturing is:. Lr =. Ly = L. E p. To obtain the labour demand for defensive activities, the de…nition of c ( quality index are used to obtain : Lf =. f h cos. 17. (20). Q. ;. ) in (8) and the average.

(19) The full employment condition requires that L = Ly + Lr + Lf , which is equivalent to: 1=. E Io + p h cos. 1. Q f + L h cos. Q L. (21). To include the free entry, the …rm value of the replacement case (15,a) is substituted on the RHS of (9) and the outsiders’R&D productivity for constant values of IoL and cos on the LHS. In addition, equation (4) must be veri…ed at. E E. = 0 , so that r = :. E=. + I0 Q p f + L p 1 h cos h cos. Clearly, in a steady-state equilibrium in which IoL and E are constant, x constant. Hence, population and average quality must grow at the same rate: L Q = =n Q L. (22) Q L. must also be. (23). Therefore, the model builds on the same tractable properties of a standard semi-endogenous growth model without scale e¤ects. After putting demands (2) into the instantaneous utility (1), taking logs and di¤erencing, the growth of the average quality implies the standard steady-state utility growth u (t) = u (t). n 1. The rate of growth of Q is obtained in the usual way. Using the law of large numbers, the variation of average quality is computed by adding the expected technological jump of each industry: R1 Q = Io [ (j! + 1) (j! )] d!. After using the de…nition of Q, this expression reduces to: 0. Q = Io Q. 1. 1. In steady state, condition (23) must hold. Thus, the innovation rate is: Io =. n 1. [. (24). 1]. Using this result and equation (18), the consequent steady-state potential menace of outsider s is: 2. 6 IoL = 6 4. 3. n [. 1. 1]. h. f. 7 7 i 5. (25). The technological bias in steady-state in the SR equilibrium is obtained by putting (25) into (17), which leads to:. 18.

(20) cos. =. ". 1. 1 n. f. #. (26). The steady-state level of bias keeps the property of the partial equilibrium decision: when ! 1 the bias vanishes (cos ! 1). By constraining the possibilities of bias, regulation is at the core of the jump from continuous …rm renewal (the SR equilibrium) to a permanent leadership (the P M one). This result is expressed in the following proposition.. Proposition 4 For > there exists a unique level of regulation de…ning the threshold between the Schumpeterian replacement (SR) and the Permanent monopolist (PM) cases involved in the value of the leader …rm as written in equation (15). Proof. See Appendix A.1.3. Figure 3 illustrates the steady-state decision of bias given the value of regulation. For 2 1; the bias is set following equation (19) within the P M equilibrium. Thereafter, the leader must take into account free entry of outsiders and decides accordingly to equation (26) in the context of the SR equilibrium.. cosξ θ 1. SR. PM. Credibility level. 1 − γ −[σ −1]. ψ. Regulation ψ. Figure 3. Leader’s optimal choice of bias The e¤ect of regulation on R&D e¤ort in steady-state can be analysed through the share of labour Lr allocated to research sr L . This share can be obtained from the system of equations (21) and (22) Q for two unknowns: x L and E . Solving this system for x and using Lr as expressed by (20) gives: 19.

(21) 1. sr = rep. Where. rep. 1+. [1. [. 1]. ]. [p 1]n. +. 1 1 [p. 1]. (27). p. +. 1 [p. 1]. : The following proposition can now be stated.. Proposition 5 In the Schumpeterian equilibrium, regulation ( ) increases the labour share allocated to R&D (sr ) and its e¤ ect is all the more important for a larger innovation size ( ) . Proof. See Appendix A.1.4. As R&D becomes harder, at equilibrium, less …rms will be willing to enter the R&D race. The aggregate labour allocated to R&D then decreases. The size of innovation increases marginal revenues as well as the cost of climbing the quality-ladder in the next R&D race. Both Schumpeterian channels combine to modulate the R&D incentives stemming from bias reductions. For p = , the multiplicative factor of in (27) is increasing in : the e¤ect of regulation is positively conditioned by the size of innovation (see the appendix for a formal proof). 3.4.2. The permanent monopolist equilibrium. In a situation with a permanent monopolist, the free entry condition in the research sector no longer holds. Instead, the steady-state equilibrium condition is given by the interest rate (14) that allows a positive and …nite amount of research. In this equilibrium some minor adaptations for labour market clearing must be considered. First, the monopolist allocate labour to research without being a¤ected by the bias. Its probability of innovative success is then IL = `L L . Second, the optimal choice of [ 1] . Full employment requires: bias is now given by cos = 1 1=. E IL + p h. 1. Q + L h 1. f [. 1]. Q L. (28). L As before, if expenditure and innovation rates are constant, then Q Q = L = n: Thus the steady-state n rate of innovation remains the same: IL = [ 1 1] . Putting the interest rate (14) into the optimal path of expenditure (4) implies:. E=. 1. [. 1]. Q p hLp 1. Since E is constant, consumption growth is still given by. u(t) u(t) = Lr L for. (29) n. 1:. The steady-state share of labour allocated to R&D srm = the permanent monopolistic case can be obtained by substituting E; as de…ned by (29), into labour market clearing (28) for IL at the steady state. This yields:. 20.

(22) srm = " Where. per. 1+. 1 per. f. + n[1. (. 1). ]. 1. #. (30). n[p 1] .. Proposition 6 In the permanent monopolist equilibrium, regulation ( ) reduces the share of labour allocated to R&D (srm ). Proof. See Appendix A.1.5. In this equilibrium the monopolist is the innovator. If regulation increases, but not enough to ensure a continuous monopolistic replacement, resources that potentially can by employed in R&D must be allocated for defensive activities. The R&D e¤ort then decreases. Proposition 4-6 can then help to explain the empirical pattern depicted in Figure 1.. 4. Conclusion. This paper has presented in a simple quality-ladders model the consequences of defensive strategies on R&D e¤ort and market structure. Among the multiplicity of available strategies, defensive reactions may increase the cost of R&D beyond the pure technological dynamics. Institutions constraining this set of strategies and reducing the resulting deterring e¤ects may increase the resources devoted to innovation. This e¤ect however is likely to be observed above a certain threshold level of regulation and should be stronger when technology jumps are larger. All these features are consistent with several patterns that emerge from data on R&D expenditure in OECD industries. Further e¤orts should be addressed to empirically provide a deeper scrutiny of the model’s mechanisms, namely with the help of …rm demographic data.. A. Appendix. A.1 A.1.1. Proofs of theoretical propositions Proof of proposition 1. The necessity of this condition comes from the fact that any credible commitment of a high R&D e¤ort depends on the capability of the leader to perform, at least, a positive amount of R&D when free entry in research is possible. Equation (11) shows that the leader …rm does perform R&D when (j! +1) vL (j! )] 1: If free entry applies, then vL (j! + 1) = o (j1! +1) = h cos : One can L [vL (j! + 1) j!. 1 o (j! ). (j! +1) h cos. [. 1]. obtain vL (j! ) by adjusting for one step down in the quality-ladder: vL (j! ) = = : j! 1 Putting these elements together and assuming that j! = yields (12). To show the su¢ ciency, observe …rst the properties of the equilibrium with permanent monopolist. Notice that because of constant returns to scale of the R&D investment, if (12) holds as an strict 21.

(23) inequality, the optimal R&D e¤ort for the leader is unbounded. If (12) holds as equality, the leader …rm can perform any …nite amount of R&D e¤ort. In both cases it can invest in R&D without taking into account outsiders menace. To show this, assume that (12) holds true. The leading position value (11) must be then evaluated for ò = 0. The only case ensuring a positive and …nite R&D investment of the leader is when:11 vL (j! + 1). 1 L (j! + 1). vL (j! ) =. (31). Putting the value of vL (j! + 1) implied by (31) into (11) (when ò = 0) yields the present optimal value of a permanent monopolist leader written in equation (13). At equilibrium, the interest rate must verify (31), otherwise the leader carries out zero R&D e¤ort or an unbounded amount. Using the monopolist pro…ts equation (6) and (13), one obtains the interest rate expressed in (14). It should be proven now that this equilibrium, obtained when the credibility condition (12) holds, is not a pro…table outcome for outsiders. Consider equation (10). The self-selection of outsiders in R&D races requires vL (j! + 1) <. 1 o (j!. (32). + 1). Using the optimal value vL (j! + 1) stated in (13), pro…ts (6) and the de…nition of (j! + 1) [ 1 where. c( ; ) p 1 EL p Q. + 1) gives:. (j! + 1) cos. <. 1]. o (j!. (33). > 0.. This self-selection condition is indeed implied by the credibility condition (12). To see it, multiply both sides of (12) by (j! + 1) > 0 to obtain: (j! + 1) [. 1 Since. (j! + 1) cos. 1]. > 0, the ful…llment of (12) then veri…es:12 (j! + 1) [ 1. 1]. <. (j! + 1) [. 1. 1]. (j! + 1) cos. (34). Thus, if the leader is credible outsiders will not carry out R&D. This establishes proposition 1. 11. Klette and Griliches (2000) in their Appendix A argue that the leader R&D investment in the closely-related model of Barro and Sala-i-Martin (1995) do not satisfy the second order condition of optimality. Here, when maximising the RHS of the corresponding Bellman equation, the leader investment is derived using a standard CRS-equilibrium rationale as, by construction, one does not deal with a concave objective fonction. 12 Notice that from (31) the equilibrium with permanent monopolist happens when the credibility condition holds as equality. Since the RHS inequality in (34) is unambigously strict, the self-selection condition will still be satis…ed.. 22.

(24) A.1.2. Proof of proposition 2. Using the …rst order condition stated in equation (16) along with Io = IoL cos de…ned by (8) gives: cos. +. j!. =. f I0L vL (j! ). and c (. j! ;. ) as. L (j! ). Recall that the free-entry condition in the previous R&D race (the one that the incumbent has 1 1 won) states: vL (j! ) = o (j = (j! ) cos . After applying this, the …rst order condition can be !) j! 1 L written as 1 f + + cos j! = cos j! 1 I0L f. De…ne now k as aj! = k. 1 +. ;. I0L. (j! ). where. (j! ) =. + j! P. < 1; aj! j. j! .. cos. The sequence of aj! = k aj!. 1. can be expressed 1. is a geometric series that converges towards. . Thus, for a. 1. j=0 1. high enough level of j! ; one has a = k 1 : Putting back the de…nitions of a; k and (17). Appropriate second-order condition requires: H 2 L (j! ) cos. +2. gives directly. <0 j!. where H IoL L (j! ) vL (j! ) cos + j! [2 + cos 2 j! ] + f [ 2 + cos 2 j! ]. By simple inspection, one notices that the second term in brackets is negative ( since > cos 2 j! ). Therefore, a su¢ cient condition for H be negative is that 2 + cos 2 j! < 0: After using the identity cos 2X = 1 2 : Given the solution of the …rst-order 2 cos X 1; observe that this happens when [1 cos 2 ] < condition, this is implicitly ensured when is su¢ ciently high. A.1.3. Proof of proposition 4. The threshold. is the one solving cos. [. = 1. 1]. . Denote. ( ). [ 1] : To prove proposition 4, it su¢ ces to show that and 1 cos 2 ]0; 1] : Taking appropriate derivative gives:. @ ( ) = @. [. ]2. Using the expression for cos ;. " @ ( ) @. 1. 1. #. f. n. ". +. ln. cos [. ]. Moreover, 23. [1. ln (cos )]. =. [. 1 n. f. 1. 1] f n. ( ) intercepts. 1. reduces to:. @ ( ) = @. cos. !#. once for.

(25) Since. >. then. cos [. ]. >0. Since cos 2 ]0; 1] then 1. ln (cos ) > 0. Thus @ @( ) > 0;which means that for > 1 and > 1 one veri…es that intercept between and : A.1.4. ( ) is a monotonically increasing function of : Furthermore, < 1: Hence, for relevant values of cos there exists a unique. Proof of proposition 5. By simple inspection of (27) one veri…es that sr ( )is increasing in p = and (27), the e¤ect of on sr is: dsr = d f. [n [. Notice that sr is a concave function of d2 sr = d 2. 2n2 [ f. n2 [ 1] 2+ 1] + ] + [n [. 1] [n [. . Analytically, using price setting. ]g2. + ]. >0. : 2+. f [n ] + [n [ 1] + ] + [n [ + ]. 1] + ]g <0 ]g3. It is easy to see that the term in braces in the numerator is positive since > : The same d2 sr remark applies to the denominator. Hence d 2 < 0 meaning that sr is concave on for any possible value of the size of innovation ( > 1) : An envelope-theorem rationale can be applied. Observing that, as increases, the function sr ( ) converges asymptotically towards its maximum sr ( ) = n[ 1] lim sr( ) = n[ . If dsdr ( ) > 0 then the shape of sr ( ) is positively a¤ected by : + +1 ]+ [ ] !1. To verify this, take partial derivatives. n f dsr ( ) = d f [n. + [n ]+. ] [1 + [ 1]]g [n [ 1] + ]g2. The sign of this expression depends crucially on the sign of the braces in the numerator. If n > 0 the ) 1 su¢ cient condition for dsdr ( ) > 0 is that G ( ) 1 [ 1] 0. Since dG( = 1 >0 d and G (1) = 0 ; G is a continuous monotonically increasing function of that is positive for any value of 2 ]1; 1[ : Hence the slope of sr ( ) increases with the size of innovation. A.1.5. Proof of proposition 6. Analysing the e¤ect of. on srm ( ) : f 1 dsrm ( ) = d n 1+. Since 0 < 1. (. 1). < 1 it follows that. dsrm d. 1) 1. ( f n. 1. (. ln 1 (. 1) 1. +. n[. 1). 1]. < 0:Thus, srm is decreasing in. 24. :.